Proposition 7

Let f be a continuous mapping of a Hausdorff space X into a locally compact space Y. Then f is proper if and only if the inverse image under f of every compact subset of Y is compact. Further, if f is proper then X is locally compact.

If f is proper and K is a compact subset of Y, then $\overline{f}^{-1}(K)$ is compact by Proposition 6 of no. 2. Conversely, if this condition is satisfied, let $(\mathbf{U}_\alpha)$ be a covering of Y by relatively compact open sets. Then the sets $\overline{f}^{-1}(\overline{\mathbf{U}}_\alpha)$ are compact in X and their interiors cover X; since X is Hausdorff this shows that X is locally compact. Furthermore, each of the mappings $f_{\overline{\mathbf{U}}_\alpha} : \overline{f}^{-1}(\alpha \overline{\mathbf{U}}) \to \overline{\mathbf{U}}_\alpha$ is proper (no. 2, Theorem 1, Corollary 2) and therefore f is proper by Proposition 3 b) of no. 1.